Almost-Exact Simulation Speeds Up Bermudan and American Option Pricing

When an option can be exercised before its expiry date, every simulated price path has to do more work. This paper tackles that problem with an almost-exact simulation scheme, or AES, for Bermudan and American options under the Heston stochastic volatility model and the more complex double Heston model. The key trick is to simulate the variance process with a non-central chi-square distribution instead of relying on a rougher approximation. The authors extend AES to the double Heston case and compare it with the Euler scheme, a common numerical baseline. Their experiments show that AES gives accurate option prices while cutting computational time, and it performs especially well for Bermudan options when the number of simulation steps matches the exercise dates. In short, the paper shows that a smarter way to generate Monte Carlo paths can make early-exercise option pricing both faster and more reliable.

Every early-exercise option asks a computer a hard question. Should it stop now, or wait one more day? Monte Carlo simulation means testing many random future paths. That works well only if each path feels believable. Here, the surprise is simple. A method called almost-exact simulation, or AES, does not slow the job down. It makes the job cleaner. It draws the market's changing risk, called volatility, from a better-shaped random rule. That helps when the option can be used before expiry. It also helps when the price follows the Heston model, a market model with changing volatility, or the tougher double Heston model.

Why the faster path matters for early exercise

The study extends AES from plain Heston markets to double Heston markets. Heston uses one changing volatility path. Double Heston uses two changing volatility paths. The same idea now prices Bermudan options and American options. Bermudan options allow exercise only on set dates. American options allow exercise on any date before expiry. The tests compare AES with the Euler scheme, a common step-by-step approximation. The result is clear. AES gives accurate prices and uses less computing time. The gain grows when the number of simulation steps matches the Bermudan exercise dates. That matters because each random path can be costly. The scheme stays robust across both models. That makes it attractive for repeated pricing runs.

How AES keeps the variance from wandering off

Variance is the part of price movement that changes over time. AES treats that part with more care. Instead of moving it with a rough guess at each step, it draws the next value from a non-central chi-square distribution. That is a probability shape for positive values. It fits the variance process better than a plain Euler step. The option value then comes from least squares Monte Carlo. That is a method that scores many simulated paths. It uses those scores to guess the best exercise time. That gives the early-exercise rule enough room to work without turning the math into a grid-heavy puzzle. The same setup works for Heston and double Heston.

The comparison the tests ran

Why faster pricing matters

That speed matters because early exercise multiplies the work. Each future path must be checked against each exercise date. A rough variance update can blur the decision. AES cuts that blur. It also gives a simpler route for the double Heston model. That model has two volatility processes. Finite difference methods solve prices on a grid. They can get awkward there. Simulation stays flexible when payoffs depend on a path or on several assets. The same machinery can then serve more complex pricing jobs. It turns the variance step from a guess into a draw with the right shape.

When the time grid slips out of sync

The surprise still holds at the end. Better-looking randomness can also make the computer work less. The sharpest next test is a Bermudan case where the time grid does not match the exercise dates. That setting would stress the very advantage AES showed here. If the same gains survive there, the method would look less like a clever tweak. It would look more like a default tool for early-exercise pricing. That would matter most for desks that price many contracts at once. It would cut time without asking for a coarser answer.